Research on low speed control position observer for accumulator electric locomotive PMSM

The PMSM of accumulator electric locomotive is generally controlled without position sensor, but the traditional position observer has the low speed observation accuracy, and the nonlinear factors of many mechanical parameters, such as torque feedback and moment of inertia, interfere with the low dynamic performance of the motor control system. A position observation model based on the nonlinear expansion observer is designed. The position observation model is constructed by using the relationship among the speed of the motor, the position angle of the rotor and its angular acceleration as well as the nonlinear function. Because it can avoid measuring the load torque and using the nonlinear expansion observer to estimate the uncertain disturbance, it is easy to realize. The simulation results show that the position observer can make the electric locomotive run stably under low speed control and has good low speed dynamic performance.


Introduction
Because the rotor position signal of accumulator electric locomotive cannot obtain accurate information from the observed back electromotive force or flux when the motor is running at low speed, high frequency injection method is often used at low speed [1].The basic idea is to superimpose the high frequency signal to the fundamental signal and inject it into the motor.The corresponding high frequency signal will carry the position information of the rotor of the motor.After signal separation processing, the information containing the position of the rotor will be obtained.For the speed and position Angle of the motor, the position observer constructed by the mechanical equation of motion is often used to obtain, but the mechanical equation of motion requires additional knowledge of the moment of inertia and torque of the motor, and the dynamic performance of this position observer is ordinary.Literature [2] studies a third-order nonlinear extended state observer algorithm for position and speed estimation of sensorless internal permanent magnet synchronous motor drivers.This method has the advantages of strong nonlinear feedback ability, fast convergence rate and strong anti-interference ability, but the construction is complicated and not suitable for the working environment of accumulator electric locomotive.Literature [3] studies a high-order phaselocked loop based on an extended observer to estimate the position of the motor rotor.This method avoids using moment of inertia and measuring the torque of the motor, and has a simple observation structure, but ignores the influence of nonlinear factors on the operation of the motor.Literature [4] studied the method combining nonlinear expanded observer and mechanical equation of motion, which realized the estimation of unknown uncertain interference and external interference when the motor ran without position sensor, but it still needed to know the mechanical parameters such as moment of inertia and the measurement of the motor running torque.Literature [5] studies a method based on the combination of extended observer and adaptive system.This method takes electrical angle, speed and load torque as state variables to construct the observer, and takes load torque as feedforward compensation in the observer to achieve faster convergence.
In order to realize the low speed position observation model of PMSM of accumulator electric locomotive running at low speed without motor torque feedback, without using any mechanical parameters, and with the function of restraining nonlinear factors from interfering the dynamic performance of the motor, this paper proposes a scheme to construct the low speed position observation model: (1) The heterodyne method was used to obtain the angle error [6][7], and the relationship among rotor position angle, angular velocity and angular acceleration was used to construct the expansion observer.(2) Considering the nonlinear factors, the extended observer is constructed as a nonlinear extended observer (Nonlinear extended observer, NESO).The simulation results show that NESO observation model can effectively satisfy the motor stable operation at low speed and restrain the interference of uncertain factors.

Position observation of PMSM running at low speed
A position observation model for low speed control of PMSM is established.Assuming that the amplitude of the injected rotating high-frequency voltage signal is V in and the frequency is ω in , the high-frequency current signal generated by rotating high-frequency voltage signal after passing through the motor contains two components, namely positive phase high-frequency current component and negative phase sequence high-frequency current component.Only the negative sequence high frequency component contains the information of the motor rotor position.By filtering out irrelevant signals, we can obtain: where: i n,αβin is the high frequency voltage component of negative phase sequence, θ e is the electrical angle.Through derivation by using heterodyne method principle, the expression of tracking error signal can be obtained: where: ˆe  is the estimated electrical angle Figure 1 shows the schematic diagram obtained by calculating the traditional rotor position tracking observer formula.In the figure, P n is the polar logarithm, T e is the electromagnetic torque, ˆL T is the estimated load torque, Ĵ is the estimated moment of inertia of the motor, K P , K i , K d are the parameters of the traditional rotor position tracking observer [8][9][10][11].
Heterodyne method PI regulator Mathematical models of mechanical systems Js K s K s K As can be seen from Equation (3), rotor position tracking observer needs to know the moment of inertia, and parameter setting is complicated, and a sensor is needed to measure the motor torque.Therefore, the position observation model is relatively limited in the low speed control system of permanent magnet synchronous motor without position sensor for accumulator electric locomotive.The nonlinear extended observer does not need to know the specific structure and system parameters of the system, so it can debug the model gain directly and select the appropriate nonlinear function, so as to achieve the convergence of the observed results.Based on the advantages of NESO, a low speed position observation model based on NESO control is established.

NESO low speed observation model
Since the derivative of the rotor's electric angle is angular velocity, and the derivative of angular velocity is angular acceleration, the three are natural series integral.The model established by taking the angular velocity differential as the expanding state variable can be applied to both the steady-state system and the transient system.
where, a e is angular acceleration and d is the derivative of acceleration.
According to the derivation of Equations ( 4), NESO is constructed: the signal passing through the heterodyne method module is taken as the input of NESO, and the estimated angular velocity and position angle is taken as the output, in which the expansion variable is the differential of angular velocity.In this way, the transformation can avoid using the mechanical equation of motion to construct the observer, as well as the measurement of motor torque.The nonlinear extended observer represents: where, β i >0(i=1,2,3), α=0.5, δ is the constant that affects the filtering effect.The purpose of using the saturation function fal(ɛ,α,δ) is to suppress the interference of the system and optimize the system. 1 , ( , , ) sgn( ), Mapping relationship of system state variables: 1 The extended state variable z 3 (t) contains the nonlinear function of unknown uncertain disturbance and internal uncertain dynamics.The angle error signal can be observed and the interference generated by nonlinear factors can be suppressed by processing in this way.When the angle error signal is large (|ɛ|>δ), the gain of the system will be reduced to prevent overshoot.When the angle error signal obtained is small (|ɛ|<=δ), the gain of the system will increase, and the convergence of the position observer will be accelerated, improving the dynamic response performance of the position observer.
Since Equation ( 5) contains nonlinear functions, it is inconvenient to analyze, so it is simplified as follows: By substituting Equation (7) into Equation ( 5), we get: where, z 1 , z 2 , z 3 is the state variable and b 1 , b 2 , b 3 is the gain of NESO Apply the Laplace transform to Equation ( 8): The NESO block diagram is set up according to Equation ( 9), as shown in Figure 2.
NESO observation error is defined: The differential form of observation error can be obtained from Equations ( 4), ( 5) and ( 11): According to Equation (12), the transfer function between speed estimation error e 2 and angular acceleration derivative d can be obtained: According to Equation ( 14), when angular acceleration d(s) tends to 0, the steady-state error of NESO observer tends to 0. When angular acceleration d(s) is 0, it means that angular acceleration, angular velocity and position angle can be tracked without static error.

The selection of NESO model gain
As the denominator of NESO transfer function expression contains s 3 term, it can be seen that the parameter setting of this model is relatively complicated.In the principle of automatic control, three same pole setting parameters of negative half plane are generally selected to achieve convergence.First, the pole -c (c>0) is selected, and the specific setting method is shown in Equation ( 15).16) and comparing the left term one by one, it can be obtained: By combining Equations ( 12), ( 16) and the final value theorem, we can get: As can be seen from Equation (17), the smaller the value of M, the larger the value of c, the smaller the steady-state error.Then, different closed-loop function poles c is designed according to Equation (10), and the following figure can be obtained.
It can be seen from Figure 3 that the larger the closed-loop pole value, the larger the closed-loop transfer function cutoff frequency.The performance requirements of NESO observer are fast response and small fluctuation of position error.Generally, the higher the system bandwidth, the better the performance of the system, but with the increase of the bandwidth, the system immunity will be reduced, so the bandwidth in a proper location is very important for the system performance.It can be seen from Equation ( 7   Three PI controller parameters are obtained through theoretical calculation, and bandwidth selection is the most important in theoretical calculation.The current loop bandwidth is related to the time constant of PMSM.For the speed loop PI controller parameters, the bandwidth is 50rad/s, and the PI controller parameters can be calculated through the motor parameters.Simulation parameter determination is based on this fine tuning.The signal processing module is mainly composed of bandpass filter and synchronous shafting high-pass filter.The purpose of the signal processing module is to obtain the negative phase sequence high-frequency current component.The heterodyne method deals with the negative phase sequence of high frequency current components to obtain the rotor position tracking error [12][13][14][15].

Simulation verification
In order to verify the feasibility of NESO position observation model, Matlab/Simulink was used to establish a PMSM control system simulation model based on NESO.The specific parameters are shown in Table 1.In the simulation experiment, the measured position and speed were compared with the estimated position and speed.Meanwhile, the simulation results based on NESO position observation model and traditional position observation model were added for comparative analysis.5(a, b) shows that the motor position tracking estimated by the traditional method is not as fast as the tracking based on NESO position observation model and when the load changes from no load to 10N•m, the traditional rotor position tracking error changes obviously, but the position tracking error based on NESO position observation model does not change basically.Figure 5(c, d) shows that given a speed of 100r/min, the traditional mode responds less quickly than the system based on NESO position observation model at startup and load mutation, and it takes longer to reach stability.
In Figure 5(e, f), when the load of the system based on NESO position observation model is increased to 40N•m, the speed and position tracking can still be done, but the traditional method cannot continue tracking.The moment of inertia of the motor has been assigned before simulation.In fact, this moment of inertia cannot be accurately obtained by traditional methods, and the sensor is required to obtain the motor torque to the position observation model.As shown in Figure 5(g, h), when the flux of PMSM is 0.2798 and decreases to 0.25, and the given torque changes to 10N•m at 1s, the robustness of the traditional observer is worse than that of the NESO observer.Therefore, the low speed control method of permanent magnet synchronous motor without position sensor based on NESO position observation model is feasible and has certain advantages.

Conclusions
In this paper, the low speed position observer based on NESO is established to be universal and does not need to measure specific parameters.By comparing with the simulation results of the traditional low speed observer, it can be found that the low speed position observer has a better control effect in the case of load mutation of accumulator electric locomotive.

Figure 1 .
Figure 1.Implementation block diagram of the rotor position tracking observer.The transfer function between the actual position of the motor rotor and the estimated position of the rotor can be obtained from Figure1:

Figure 2 .
Figure 2. Implementation block diagram of NESO tracking observer.According to Figure 2, the transfer function between the estimated rotor position ˆ( ) e s  and the actual rotor position ( ) e s  can be obtained: Let d(s)=M/S, M is a constant, according to the final value theorem: decomposing the right term of Equation ( ) that b 1 , b 2 and b 3 are nonlinear functional relations, so the calculated gains β 1 , β 2 and β 3 are only approximate values.

6
By deducing the principle of NESO and setting the gain, the NESO position observation model is established, and NESO replaces the traditional position observation model, as shown in Figure4.

Figure 4 .
Figure 4. Block diagram of NESO control system.